Diffusion-Limited Kinetics of Isovalent Cation Exchange in III–V Nanocrystals Dispersed in Molten Salt Reaction Media

The goal of this work is to determine the kinetic factors that govern isovalent cation exchange in III–V colloidal quantum dots using molten salts as the solvent and cation source. We focus on the reactions of InP + GaI3→ In1–xGaxP and InAs + GaI3→ In1–xGaxAs to create technologically important ternary III–V phases. We find that the molten salt reaction medium causes the transformation of nearly spherical InP nanocrystals to tetrahedron-shaped In1–xGaxP nanocrystals. Furthermore, we determine that the activation energy for the cation exchange reaction is 0.9 eV for incorporation of Ga into InP and 1.2 eV for incorporation of Ga into InAs, both much lower than the measured values in bulk semiconductors. Next, we use powder XRD simulations to constrain our understanding of the structure of the In1–xGaxP nanocrystals. Together our results reveal several important features of molten salt-mediated cation exchange and provide guidance for future development of these materials.

S4 nitrogen. A portion of (TMS)3P (375 mg, 2 mmol) was added to the InCl3 solution, and the reaction was heated at 270 °C for 30 hours. The solution was cooled down to room temperature, another portion of (TMS)3P (375 mg, 2 mmol) was added to the precursor mixture stored previously and that was added to the reaction flask. It was heated at 280 °C for another 30 hours and finally cooled down. The QDs were washed in a nitrogen glove box three times by solvent/nonsolvent (toluene/EtOH) precipitation and centrifugation. As synthesized, these InP QDs are quite polydisperse, but the size distribution could be narrowed through size-selective precipitation.
Fractions of successively smaller particles were isolated by slow addition of EtOH and centrifugation. The QDs were stored as a solution in toluene. The predominant fraction of the largest particles was used for this work.
Large, tetrahedral InP QDs. Large, tetrahedral InP QDs were synthesized based on an established procedure. [3][4] In a 100 mL 3-neck round bottom flask, 442 mg InCl3 (2 mmol) was dissolved in 10 mL oleylamine and degassed under dynamic vacuum at 120 °C for an hour. The solution was heated to 260 °C under nitrogen, and 1.3 mL (NMe2)3P (7.1 mmol) was quickly injected. The reaction proceeded at 260 °C for 30 minutes and was then cooled to room temperature naturally.
The particles were washed three times by solvent/nonsolvent (toluene/IPA) precipitation and centrifugation. A narrower size distribution was achieved by size-selective precipitation, where EtOH nonsolvent was slowly added to a solution of QDs in toluene and successively smaller particles were isolated by centrifugation. The QDs were stored as a solution in toluene.
InAs nanocrystal synthesis: InAs nanocrystals were synthesized using InCl and tris(dimethylamino)arsine. 5 Briefly, 75ml of OAM was loaded into a 250ml European-style flask and degassed under dynamic vacuum at 110°C for several hours. The flask was cooled to room temperature, and 2.25g of InCl and 3ml of TOP were added under nitrogen flow. The reaction S5 mixture was heated to 240°C. Separately in a glovebox, 12ml OAM and 1.125ml of DMA-As were mixed and heated to 50°C until bubbles stopped forming. 4ml of this solution was loaded into a syringe and the remaining 8ml was loaded into a second syringe. The solution from the first syringe was injected into the InCl/OAM mixture at 240°C and the reaction was allowed to proceed for 20 min. Next, the remaining 8ml of OAM/DMA-As solution in the second syringe was injected at a rate of 4ml/hr. After the end of the injection, the reaction mixture was heated to 280°C for 20 min. Nanocrystals were recovered from the solution in a N2 glovebox using anhydrous ethanol as a nonsolvent and toluene as a solvent respectively.
Ligand exchange: Native organic ligands on the InAs and InP nanocrystals were replaced with GaI3 using methods adapted from Dirin et.al. [6][7] Briefly in a N2 glovebox, 5ml of 0.05M GaI3 solution in DMF was prepared in a 20ml vial and 10ml of hexane was added to create a biphasic system. The nanocrystals dissolved in hexane were added to the top phase, and the biphasic mixture was stirred for several hours until phase transfer of the nanocrystals to the lower DMF phase was complete. The DMF phase was decanted and washed three times with excess hexane.
Next, the nanocrystals were precipitated using ACN, collected by centrifugation, and dispersed again in DMF. The nanocrystals were then precipitated a second time using ACN and collected as a pellet via centrifugation. This pellet was then washed three times with excess ACN and dried overnight in a glovebox.

Molten salt dispersion:
In a N2 glovebox, GaI3 and KI in either 1:1 or 65:35 molar ratio were loaded into an oven-dried mortar and pestle and ground into a fine powder. For InP nanocrystals, the previously prepared dry powder of ligand exchanged nanocrystals was added to the mortar and pestle with the eutectic mixture of GaI3 and KI in 65:35 molar ratio and ground to disperse the powder in the salt mixture. Next the nanocrystal/salt mixtures were weighed out into oven dried S6 quartz ampoules (9mm outer diameter 7mm inner diameter, with an indentation ~50mm from the bottom), and a 6mm diameter quartz plug was inserted. The samples were heated at 240 o C for 1 hour to melt the salt mixture. After melting, the ampoule was sealed using a H2/O2 torch under vacuum using an air free vacuum transfer chuck to mount the ampoule on a vacuum manifold without air exposure. The procedure was the same for InAs nanocrystals, except a KI/GaI3 salt mixture in 1:1 molar ratio was used. It was melted together before adding the nanocrystals, allowed to solidify, and ground again into a fine power.
High temperature annealing: For InP samples, nanocrystals were annealed in a custom-built shaking furnace. The furnace consisted of ceramic walls around a central ceramic tube to hold the ampoule. The furnace was heated by a swaged heater cartridge with the temperature measured using a thermocouple and controlled using a PID temperature controller. The furnace assembly was placed atop a small vortex shaker. For InAs nanocrystals, the samples were annealed in a custom high temperature heating block, which consisted of a 4-inch diameter cylindrical aluminum block with 8, 3/8in diameter holes drilled in a circle around a central thermocouple hole. The block was wrapped in high temperature heating tape (BriskHeat BWH) and several layers of insulation.
The temperature was controlled using a PID temperature controller. The temperature in each hole was independently measured to be within ±2°C of the set temperature.  Transmission electron microscopy. TEM images were obtained on an FEI Tecnai F30 microscope at 300 kV. For measurement of the gallium content distribution, special care was taken S8 to minimize microscope variations which would affect the magnification. All data was collected during the same TEM session. The illumination conditions were set such that the convergence angle of the illumination beam was identical for each sample by using a fixed "spot size" and C2 lens excitation. Next, the sample height was placed within 500 nm of eucentric focus using the stage height adjustment and all data was collected while maintaining the focus of the objective lens within ±500nm of the eucentric focus to minimize focus dependent magnification effects.

Small-angle x-ray scattering (SAXS).
Colloidal solutions of QDs in toluene were prepared in sealed Kapton capillaries for small-angle x-ray scattering (SAXS) experiments. SAXS patterns were collected using a SAXSLab Ganesha instrument with Cu Kα radiation (λ = 1.54 Å). The SAXS curves were analyzed by fitting to a quantitative model in Igor Pro using the Irena package (available at http://usaxs.xray.aps.anl.gov/staff/ilavsky/irena.html). 8 The scattering curves were fit in the particle size distribution module using the model-free Maximum Entropy approach. Based on TEM data, the particles' form factor was assumed to be that of a sphere with an aspect ratio of 1. The extracted size distributions were further fit with symmetric Gaussians to estimate average particle diameters along with the associated standard deviations.
Powder x-ray diffraction simulations. Powder X-ray diffraction patterns were simulated from structures built using the Atomsk software package. 9 Briefly, large slabs of InP, GaP, and In1-xGaxP were generated by first preparing InP slabs, but with a lattice parameter which corresponded to the desired alloy gallium content. Next, a percentage of In atoms were randomly selected and converted to Ga atoms to generate alloy structures with the appropriate lattice constant and gallium composition. A spherical cutoff of desired diameter was used to generate nanocrystal structures.
A similar approach was used to prepare In1-xGaxAs nanocrystal structures. Atomic structures were visualized using the VESTA software package. 10 X-ray diffraction patterns were simulated using S9 the DebyeByPy software package 11 which calculates intensities (I) as a function of the scattering vector (q) in units of Å −1 using the Debye formula: where the sums are over all of the atoms, f is the angle-dependent scattering factor calculated by Hartree-Fock wavefunctions by Cromer and Mann, 12 and r is the distance between atoms in Å. 13 Experimental data and simulated data was plotted either in q or in 2θ (2 = 2 arcsin( ), where is wavelength of the X-ray source).

First-principles calculations initialization.
For structural relaxation calculations, we determined the initial geometries of the nanoparticles studied here by cutting out different sized spherical nanoparticles from three dimensional zinc blende crystal structures using the Nanocut software. [14][15] To saturate the surface dangling bonds, fictitious hydrogen atoms with fractional charges were used. Each In atom is surrounded by four P(Ga) atoms, and the formal valence of the In atom is 3 and that of a P(Ga) atom is 5. Nominally, 0.75 electron of each In atom and 1.25 electron of each P(Ga) atom contribute to the formation of a In-P(Ga) covalent bond. In all the calculations, fictitious hydrogen atoms with charges of 0.75 electron and 1.25 electron were chosen to saturate In and P(Ga) atoms, respectively, so as to avoid the presence of any dangling bonds at the surfaces.

Supplementary Discussion on XRD simulations
We aim to understand how other structural changes (e.g., shape, core/shell structure, etc.) influence the expected PXRD patterns for In1−xGaxP nanocrystals. We observe that the (002) peak intensity is modulated by the particle shape. Tetrahedron-shaped particles give a broader (002) peak compared to spherical particles of the same volume; however, the integrated area of the peak appears unaffected by the shape ( Figure S12). This indicates that stacking disorder rather than the observed shape changes is likely the main factor attenuating the (002) peak intensity in our nanocrystal samples. Finally, we consider the partitioning of the gallium and indium atoms in the nanocrystals. We simulated PXRD patterns for In0.50Ga0.50P structures with random alloy and core/shell geometries ( Figure S13). Here we find that the calculated XRD pattern is minimally affected by the distribution of the gallium and indium, with only miniscule changes to the width of the (002) peak for inhomogeneous elemental distributions. Further, accounting for strain in the S17 core shell structure leads to minimal changes to the observed PXRD pattern ( Figure S14). Based on comparison of our experimental PXRD patterns and the simulations, we can conclude that our In1−xGaxP nanocrystals have considerable stacking disorder, and that gallium is incorporated based on the peak shifts, but PXRD does not provide adequate observing power to discern minute changes to the particle shape or the elemental distribution within a single nanocrystal. More advanced X-ray or TEM based techniques will likely be needed to understand these structural parameters.

HRTEM Composition Variation Analysis
Elucidating the distribution of gallium content among an ensemble of In1−xGaxP nanocrystals remains an important problem. To this end, we collect HRTEM images of InP and In1−xGaxP nanocrystals using identical imaging conditions (see methods). From these we calculate the lattice constant for ~120 individual InP and In1−xGaxP nanocrystals using an automated MATLAB routine (see methods and Figure S15). A probability density histogram of the measured lattice parameters for an ensemble of individual InP nanocrystals is shown in Figure S18A. A Gaussian fit to the histogram returns a distribution centered at 5.89Å, in good agreement with bulk InP, and a standard deviation of 0.057Å. The standard deviation of the lattice parameter measurement for the InP nanocrystals (which should have nearly identical lattice constants) represents a measure of the S22 precision of the HRTEM measurement approach we have used. We can thus ascribe any increase in the measured distribution width to real increases in the measured lattice parameter variation.
We note that microscope focus variations and tilting the crystals off zone axis (which could vary sample-to-sample) appear to have negligible effects on the measured lattice parameter for these nanocrystals as suggested by multislice HRTEM image simulations ( Figure S16). We performed identical measurements on a sample of In1−xGaxP nanocrystals and found a lattice constant distribution centered at 5.76Å with a standard deviation of 0.088Å ( Figure S18B). This indicates there is a measurable increase in the width of the lattice constant distribution upon incorporation of Ga into InP nanocrystals.
One possible explanation for observed broadening is that the size dependent kinetics of gallium incorporation observed in the previous section cause the smaller nanocrystals in the ensemble to incorporate more gallium than the larger nanocrystals in the ensemble. As a result, we would expect to observe a correlation between particle size and measured lattice parameter where smaller particles have a smaller lattice parameter. To this end, we developed an automated approach to measure the size of each particle using HRTEM ( Figure S17). For InP we observe a weak positive correlation between particle size and measured lattice constant. Upon incorporation of Ga, we observe the slope of this correlation meaningfully increases, indicating the gallium incorporation is nonuniform among the In1−xGaxP particles depending on their size. This points out an important practical consideration for preparing In1−xGaxP nanocrystals with narrow emission linewidths. First, starting with an InP sample with narrow size distribution is critical, since disproportionate incorporation of gallium into the smaller particles serves to blue shift their emission, which is already on the blue edge of the emission spectrum. Second, reaction conditions must be developed which minimize size dependent gallium incorporation. For example, S23 developing cation exchange conditions through which the extent of gallium incorporation is determined by thermodynamics rather than through diffusion kinetics would allow for a steady state condition to be achieved, thus minimizing size dependent gallium incorporation. This endeavor likely will need greater precision for the single particle lattice parameter measurement, which will be possible using modern highly stable TEMs and appropriate additional data analysis considerations. 18 For example in our analysis we used any particle which displayed (111) lattice fringes for determining the lattice parameter. In previous work using this approach on metal nanocrystals, 18 a key data analysis requirement was to only measure particles which were on a well defined zone axis such that the crystal tilt could be precisely known and thus even small errors in lattice parameter determination from crystal tilt could be avoided. At present it was not feasible to collect enough data such that reasonable statistics of particles on zone axis could be obtained.
Advances in automated TEM data collection may enable this in near future. (111) peaks are identified using the fast peak find algorithm 19 in MATLAB. Next, the peak positions were refined by fitting the peak position with subpixel accuracy (D) by using a Taylor expansion of a 3 by 3 window around the maximum intensity pixels and determining where the derivative would be zero. The (111) spacing was measured as the distance between the two (111) reflections in the FFT. The lattice parameter was calculated assuming the particles had an ideal cubic zinc blende phase. Only particles which had (111) planes as the most intense peaks were used. Particles rarely had more intense (100) or (110) peaks, and in these cases appropriate distance cutoff filters were applied to remove those cases. (D) Histogram of the measured lattice parameter for 15 tilt conditions ranging from 0º around the S25 X and Y axes, and 10º around both X and Y axes simultaneously. (E) Histogram of the measured lattice parameter of 6 defocus conditions for a particle tilted off the (110)   image of an InP nanocrystals followed by computing the FFT of the image (B). Next a mask is S26 applied to isolate the signal from the (111) lattice fringes (C) followed by computing the inverse FFT (D). Next the Fourier filtered image is binarized (E) with a threshold which only keeps the (111) lattice fringes. Finally, this image is dilated to fill in the dark space between the lattice fringes (F) and the area of the particle is determined. We quantify particle size as the diameter of a circle which would have equal area to the measured irregular region. We note that this measurement is prone to several sources of error. First, the quality of the lattice fringes is different image-to-image, and thus it is difficult to reliably define an intensity cutoff for the binarization step. For these reasons, there is a large spread in the area determinations compared to the size distribution of the sample measured by conventional TEM methods; however, on average particles which are larger give larger areas as determined by the automated method used here. For these reasons, we say our results are semi-quantitative and serve to illustrate the principle of size dependent gallium incorporation but may not provide a precise or accurate measure of the degree of this effect. The mathematical theory of diffusion in isotropic substances is based on the hypothesis that the rate of transfer of diffusing substance through unit area of a section is proportional to the concentration gradient measured normal to the section, i.e.

= − / … (1)
where F is the rate of transfer per unit area of section, C the concentration of diffusing substance, x the space coordinate measured normal to the section, and D is called the diffusion coefficient. If the diffusion coefficient is constant, the rate of diffusion reduces to

S28
In most cases ( ) is either zero or constant; therefore, the integral in equation (3) can be evaluated readily. Now, consider a spherical particle with radial coordinate r. When the diffusion is only in the radial direction, the diffusion equation (2)